Leeds wasted no time in moving onto the front foot and it required a fine save from Jack Butland to deny the home side an early goal, with the England stopper reacting well to keep out Kemar Roofe's volley at the near post.
The home side remained on the front foot and got the goal that their play deserved in the 15th minute through Klich, who converted into the bottom corner from eight yards out after Samuel Saiz had drifted past two defenders before playing a precise through-ball to the Pole.
Stoke had been second best throughout the opening quarter, but Tom Ince almost got the Potters back on level terms with a speculative effort from 35 yards which struck the crossbar and went behind for a goal kick.
James McClean should have done better when granted with a two-on-one scenario, and that escape appeared to re-energise Leeds who went on to dominate the remainder of the first half.
Ezgjan Alioski saw a volley brilliantly saved by Butland from 12 yards out and Roofe dragged a low effort agonisingly wide of the far post, but Leeds finally had their second in added-on time.
The ball was worked to Hernandez on the edge of the penalty area and despite Butland getting down well to get his hands in the way of the Spaniard's 18-yard effort, his shot was fumbled into the bottom corner.
Stoke needed half time to regroup and, seven minutes into the second half, the Potters were gifted a way back into the match as Barry Douglas hauled down Ince in the penalty area, giving Benik Afobe the chance to convert from 12 yards.
However, Douglas quickly redeemed himself by providing the assist for Leeds' third goal, with the Scot sending an inswinging corner into the area for Cooper to glance a header into the far corner.
The visitors needed to find another way back into the contest but it failed to come, and Leeds should have added a fourth only for Klich to scuff a close-range effort well wide of Butland's near post.
Stoke introduced Bojan Krkic and Peter Crouch into the action but the Staffordshire-based outfit went into the final 10 minutes with only a free kick from McClean forcing a save out of Bailey Peacock-Farrell.
The young goalkeeper also had to remain alert to keep out a close-range effort from Joe Allen, but the home side comfortably held on to seal the perfect start for Bielsa.
LEEDS UNITED (4-2-3-1): Peacock-Farrell, Ayling, Berardi, Cooper, Douglas; Phillips, Klich (Dallas 76'); Alioski (Harrison 89'), Saiz (Baker 88'), Hernandez; Roofe
STOKE CITY (4-3-3): Butland; Bauer, Martins Indi, Shawcross, Pieters; Allen, Ndiaye (Krkic 63'), Etebo (Fletcher 63'); Ince, McClean, Afobe (Crouch 76')